Why Momentum Factor Loadings Change with the Market Benchmark

Summary

The document asks why the estimated momentum coefficient in a Carhart four-factor analysis of mutual fund returns changes when the market benchmark is switched. The sample consists of 1,000 funds over 2000–2017. The reported momentum estimates are negative under both benchmarks, but differ in magnitude: −0.06 with the CRSP total market index and −0.01 with the Wilshire 5000. Both estimates are reported as statistically different from zero.

The explanation uses the Frisch–Waugh–Lovell interpretation of multiple regression. A factor coefficient represents the relationship between that factor and returns after removing the variation each shares with the other regressors. Changing the market index changes what market-related variation is controlled for, and therefore changes the residual variation used to estimate the momentum loading. The difference can arise from how each benchmark co-varies with fund returns and momentum. The document offers a conceptual explanation, not a determination of which benchmark is more appropriate or a full robustness analysis.

Key ideas

• A multiple-regression factor coefficient measures a relationship after controlling for the other included regressors.
• Changing the market benchmark changes the variation removed from returns and the momentum factor.
• Different benchmark co-movements can therefore produce different momentum loadings.
• The reported estimates are both negative and statistically distinguishable from zero, but have different magnitudes.

Tags

# Carhart (1997) momentum factor loading


# Carhart (1997) momentum factor loading












I am evaluating the performance of a sample of 1000 mutual funds over the period 2000 to 2017 using Carhart (1997) four factor model.

As a way to test for robustness, I use two benchmarks. The CRSP total market index (from Dr Kenneth French website) and Wilshire 5000.

I ran individual time series regressions for all 1000 funds. Then I cross-sectionally average the coefficients to get a sense of how my sample performs on aggregate.

When I use CRSP market index as a benchmark, I get a momentum coefficient of -0.06.

However, when I use Wilshire 5000 as a benchmark, I get a momentum coefficient of -0.01.

Both coefficients are statistically different from 0.

My question is, what could be a reason why momentum differs significantly when I change the benchmark?

Thank you.

## Answer by jd8 (score 0, accepted)

https://quant.stackexchange.com/a/35503

See this wiki Frisch-Waugh-Lovell Theorem

and it will explain how to interpret multiple regression coefficients from a regression of the form \begin{equation} Y = X_1 \beta_1 + X_2 \beta_2 + u \end{equation} as capturing the relationship between some variable $X_2$ and $Y$ after the relationship between $X_2$ and $X_1$ and the relationship between $Y$ and $X_1$ has been removed (we only consider residual correlation after controlling for other variables).

Imagine now that you have a regression where you change $X_1$ from your CRSP index to the Wilshire 5000 as you keep $X_2$ as your momentum index - the change must come from co-variation of these indices with returns or momentum.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.